Truels, or the survival of the weakest

Abstract

In this paper we review some of the main results obtained in the field of truels. A "truel" is a generalization of a duel involving three players. Depending on the rules used for chosing the players, we may distinguish between the random, sequential and simultaneous truel. A paradoxical result appears in these games, as the player with the highest marksmanship does not necessarily possess the highest survival (or winning) probability. In this work we limit ourselves to the random and sequential truels in which players use their best possible strategy with no coalitions. Furthermore, we have modified the random truel and converted it into an opinion model. In this version each of the three players holds a different opinion on a given topic. We address next the question of who wins a "truel league". We will see that, despite the paradoxical result mentioned above, still the distribution of winners is peaked around the players with the higher marksmanship for the random and opinion versions. In the sequential truel, however, the paradoxical result remains partially since the distribution of winners is peaked around the intermediate players. If the rules of truels are extended from three to $N$ players, the paradoxical results shows up even more clearly since as $N$ increases it is more difficult for the player with the highest marksmanship to win the game. Finally, we consider the dynamics of the games in a spatial distribution in a given network of interactions.